Question: Select all polynomials that have $(x+2)$ as a factor. Choose all answers that apply: Choose all answers that apply: (Choice A) A $A(x)=x^3-3x^2-10x$ (Choice B) B $B(x)=x^3+5x^2+4x$ (Choice C) C $C(x)=x^3-2x^2-13x-10$ (Choice D) D $D(x)=x^3-6x^2+11x-6$
Answer: The following statements are equivalent: $(x+2)$ is a factor of $p(x)$ $p(x)$ is divisible by $(x+2)$ The remainder of $\dfrac{p(x)}{x+2}$ is $0$ We can use the polynomial remainder theorem to solve this problem: For a polynomial $p(x)$ and a number $a$, the remainder on division by $x-a$ is $p(a)$. According to the theorem, the remainder when $p(x)$ is divided by $(x+2)$, which can be rewritten as $(x-({-2}))$, is equal to $p({-2})$. So to check each polynomial if it has $(x+2)$ as a factor, we need to check if that polynomial's value at ${x=-2}$ is zero. $\begin{aligned} A({-2})&=0 \\\\ B({-2})&=4 \\\\ C({-2})&=0 \\\\ D({-2})&=-60 \end{aligned}$ In conclusion, the following polynomials have $(x+2)$ as a factor: $A(x)=x^3-3x^2-10x$ $C(x)=x^3-2x^2-13x-10$